Automorphisms of free groups. I — erratum

New York Journal of Mathematics

New York J. Math. 22(2016) 1135–1137.

Automorphisms of free groups. I — erratum

Laurent Bartholdi

Abstract. I report an error in Theorem A of Automorphisms of free groups. I, New York J. Math. 19(2013), 395–421, where it was claimed that two filtrations of the group of IA automorphisms of a free group coincide up to torsion.

In fact, using a recent result by Day and Putman, I show that, for a free group of rank 3, the opposite conclusion holds, namely that the two series differ rationally.

1. Introduction

Let F denote a free group of rank r. Filter F by its lower central series (Fn)n≥1, defined by F1 = F and Fn = [F, Fn−1]. Let A denote the au- tomorphism group of F, and let A₁ denote the kernel of the natural map A → GLr(Z) = Aut(F/F⁰). The group A1 has two natural filtrations: on the one hand, its lower central series, defined as above by γ1 = A1 and γ_n = [A₁, γn−1], and on the other handA_n = ker(A₁ Aut(F/F_n+1)). We have γn≤An for alln.

Andreadakis conjectures [1, page 253] thatA_n=γ_n, and proves his asser- tion for r = 3, n≤3 and for r= 2. This is further developed by Pettet [7], who proves that γ₃ has finite index in A₃ for all r, building her work on Johnson’s homomorphism [6].

It was noted in [3, Theorem A] that, if r = 3, the groups γ7 and A7

differ, disproving Andreadakis’s conjecture. It was however also erroneously claimed there that An/γn is a finite group for all n. The “proof” relied on the unproven assertion that the filtrations (γ_n)n≥1 and (A_n)n≥1 define the same topology on A1. Theorem A should be replaced by the following statement:

Theorem A. The filtrations(γn)n≥1 and(An)n≥1 differ rationally atn= 4 for r = 3, and we have

(A₄/γ₄)⊗Q∼=Q³.

Received September 20, 2016.

2010Mathematics Subject Classification. 20E36, 20F28, 20E05, 20F40.

Key words and phrases. Lie algebra; Automorphism groups; Lower central series.

ISSN 1076-9803/2016

1135

1136 LAURENT BARTHOLDI

Let Ac1 = proj limA1/An denote the completion ofA1 under the filtration (An)n≥1, let (cγn)n≥1 denote its closed lower central series, and let (Acn)n≥1

denote the closure of (An)n≥1 in Ac1. Then Ac7/γb7 ∼=Z/3.

Proof. In [8], Day and Putman give explicit presentations ofA1for allr, by generators, relators and endomorphisms (see [2]). Here is a small adaptation of their result. LetE be the free group generated by the set

S:={M_i,[j,k]: 1≤j6=i6=k≤r, j < k} ∪ {C_i,j : 1≤i6=j≤r}.

These are the Magnus generators of A1, and act on F respectively by x_i 7→x_i[x_j, x_k] and x_i 7→x^x_i^j,

all other generators being fixed.

Day and Putman give explicit finite setsR ⊂E⁰ (of size around 30) and Θ⊂End(E) (of size around 4) such that

A1 ∼=hS |w^θ for allw∈R and allθ∈Θ^∗i.

Furthermore, Θ induces automorphisms ofA1 that generate the conjugation action of Aut(F) on A₁.

Using the algorithm described in [4], implemented in [5], it is possible to compute nilpotents quotients of A₁ of arbitrary class. I entered Day and Putman’s presentation inGapforr= 3, and computed (in about 1 minute) its class-4 quotient. The result, atop the calculations in [3] gives (with a^b for (Z/aZ)^b)

n= 1 2 3 4

γn/γn+1 Z⁹ Z¹⁸ Z⁴³×2¹⁴×3⁹ Z¹²³×2⁵⁰×4³×8³×3⁴⁵×9⁹ A_n/A_n+1 Z⁹ Z¹⁸ Z⁴³ Z¹²⁰

.

We deduce A₅/γ₅∼=Z³×torsion.

For the second claim, it suffices to note that the computer calculations described in [3] actually manipulate (approximations of) the groupAc1rather

thanA1.

Acknowledgments

I am grateful to Matt Day and Andy Putman for their generous insights, discussions and patience in resolving the discrepancy between their work and the original Theorem A.

References

[1] Andreadakis, Stylianos. On the automorphisms of free groups and free nilpotent groups.Proc. London Math. Soc.(3)15(1965), 239–268,MR0188307(32 #5746),Zbl 0135.04502.

[2] Bartholdi, Laurent. Endomorphic presentations of branch groups. J. Alge- bra 268 (2003), no. 2, 419–443. MR2009317 (2004h:20044), Zbl 1044.20015, arXiv:math/0007062, doi:10.1016/S0021-8693(03)00268-0.

REFERENCES 1137

[3] Bartholdi, Laurent.Automorphisms of free groups. I.New York J. Math19(2013), 395–421.MR3084710,Zbl 1288.20039,arXiv:math/1304.0498.

[4] Bartholdi, Laurent; Eick, Bettina; Hartung, Ren´e. A nilpotent quotient algo- rithm for certain infinitely presented groups and its applications,Internat. J. Algebra Comput. 18 (2008), no. 8, 1321–1344. MR2483125 (2010h:20002), Zbl 1173.20023, arXiv:0706.3131, doi:10.1142/S0218196708004871.

[5] Hartung, Ren´e. LPRES — L-Presented Groups, Version 0.3.0. 2016. http://

laurentbartholdi.github.io/lpres/.

[6] Johnson, Dennis. An abelian quotient of the mapping class group Ig.Math. Ann.

249(1980), no. 3, 225–242.MR579103(82a:57008),Zbl 0409.57009.

[7] Pettet, Alexandra. The Johnson homomorphism and the second cohomology of IAn. Algebr. Geom. Topol. 5 (2005), 725–740. MR2153110 (2006j:20050), Zbl 1085.20016,arXiv:math/0501053.

[8] Day, Matthew B.; Putman, Andrew. On the second homology group of the Torelli subgroup of Aut(Fn).arXiv:1408.6242.

[email protected]

http://www.uni-math.gwdg.de/laurent

(Laurent Bartholdi)D´epartement de Math´ematiques et Applications, ´Ecole Nor- male Sup´erieure, Paris and Mathematisches Institut, Georg-August Univer- sit¨at zu G¨ottingen

This paper is available via http://nyjm.albany.edu/j/2016/22-52.html.